Exponential Growth / Decay Calculator
Model exponential growth or decay, population, investment, radioactive decay, viral spread, or any exponential process.
Exponential growth is one of the most counterintuitive forces in nature and finance. The first half of the doubling periods feels slow; the second half feels explosive. This calculator makes that visible, seeing the period-by-period table shows why compound interest and viral spread both feel like nothing is happening right up until the moment everything is happening.
The rule of 72: divide 72 by the growth rate to find the doubling time. At 8% per year, an investment doubles in about 9 years (72 ÷ 8). At 24% annual interest rate on debt, the balance doubles in 3 years.
The exponential growth formula
The exponential growth formula is A = P × (1 + r)^t, where P is the initial value, r is the rate per period, and t is the number of periods. For decay, r is negative (or use the decay formula A = P × (1 − r)^t). This is the same formula as compound interest, exponential growth and compound interest are identical mathematical processes.
Doubling time and the rule of 72
The doubling time is the number of periods required for a quantity to double. The exact formula is ln(2) ÷ ln(1 + r). The rule of 72 approximates this as 72 ÷ (rate as percentage). At 6% growth: exact doubling time is 11.9 years; rule of 72 gives 12 years. The approximation is accurate enough for planning within a few tenths of a period.
Half-life for decay
For exponential decay, the half-life is the time for a quantity to reduce by half. Radioactive decay is the classic example, carbon-14 has a half-life of 5,730 years, meaning after 5,730 years, half the carbon-14 atoms remain. Drug elimination from the body follows first-order kinetics with a half-life that varies by drug. The half-life formula: ln(2) ÷ |r|.
Practical applications
Exponential models appear in: compound interest (finance), population growth (biology and demography), radioactive decay (physics), viral spread (epidemiology), bacterial growth (microbiology), technology adoption curves, Moore's Law in computing, and learning curves in manufacturing. Any process where the rate of change is proportional to the current quantity follows exponential dynamics.
Frequently asked questions
What's the difference between exponential and linear growth?
Linear growth adds the same amount each period (e.g., saving $100/month, after 10 months you have $1,000). Exponential growth multiplies by the same factor each period (e.g., 8% annual return, after 10 years $1,000 becomes $2,159). Linear growth is additive; exponential is multiplicative. The exponential dominates at large t.
Can anything grow exponentially forever?
In practice, no, resource constraints, saturation, and negative feedback eventually slow exponential growth. Population growth slows as resources become scarce. Viral spread slows as susceptible individuals decrease. Financial markets can't grow exponentially indefinitely because there are limits to the underlying economy. Real systems exhibit sigmoidal (S-curve) growth: exponential in early stages, then slowing to a plateau.
What is continuous compounding?
Standard compound interest compounds at discrete intervals (monthly, annually). Continuous compounding uses the natural exponential: A = P × e^(rt), where e ≈ 2.718. At 8% continuous compounding for 10 years: $1,000 × e^(0.8) = $2,226, slightly higher than discrete annual compounding ($2,159). The difference between continuous and daily compounding is negligible for most purposes.